Outlines

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National Taiwan Ocean University MSVLAB
(?????) Department of Harbor and River
Engineering
Scattering of flexural wave in thin plate with
multiple holes by using the null-field integral
equation method
Wei-Ming Lee1, Jeng-Tzong Chen2 Ching-Lun Chien1,
Yung-Cheng Wang1 1 Department of Mechanical
Engineering, China Institute of Technology,
Taipei, Taiwan 2 Department of Harbor and River
Engineering, National Taiwan Ocean University,
Keelung, Taiwan
2008?05?14? ??????
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Outlines
1. Introduction
2. Methods of solution
3. Illustrated examples
4. Concluding remarks
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Outlines
1. Introduction
2. Methods of solution
3. Illustrated examples
4. Concluding remarks
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Introduction

• Circular holes can reduce the weight of the whole
  structure or to increase the range of inspection.
  
• Geometric discontinuities result in the stress
  concentration, which reduce the load carrying
  capacity.
• The deformation and corresponding stresses
  produced by the dynamic force are propagated
  through the structure in the form of waves.

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Scattering

• At the irregular interface of different media,
  stress wave reflects in all directions
  scattering
• The scattering of the stress wave results in the
  dynamic stress concentration

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Overview of numerical methods
Domain
Boundary
IE
MFS,Trefftz method MLS, EFG
PDE- variational
DE
? ?
? ?
??
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Literature review

• From literature reviews, few papers have been
  published to date reporting the scattering of
  flexural wave in plate with more than one hole.
• Kobayashi and Nishimura pointed out that the
  integral equation method (BIEM) seems to be most
  effective for two-dimensional steady-state
  flexural wave.
• Improper integrals on the boundary should be
  handled particularly when the BEM or BIEM is
  used.

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Motivation
Numerical methods for engineering problems
FDM / FEM / BEM / BIEM / Meshless method
BEM / BIEM
Treatment of singularity and hypersingularity
Boundary-layer effect
Ill-posed model
Convergence rate
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Objective

• For the plate problem, it is more difficult to
  calculate the principal values
• Our objective is to develop a semi-analytical
  approach to solve the scattering problem of
  flexural waves and dynamic moment concentration
  factors in an infinite thin plate with multiple
  circular holes by using the null-field integral
  formulation in conjunction with degenerate
  kernels and Fourier series.

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Outlines
1. Introduction
2. Methods of solution
3. Illustrated examples
4. Concluding remarks
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Flexural wave of plate
Governing Equation
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Problem Statement
Problem statement for an infinite plate with
multiple circular holes subject to an incident
flexural wave
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The integral representation for the plate problem
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Kernel function
The kernel function is the
fundamental solution which satisfies

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The slope, moment and effective shear operators
slope
moment
effective shear
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Kernel functions
In the polar coordinate of
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Direct boundary integral equations
displacement
with respect to the field point x
slope
with respect to the field point x
normal moment
with respect to the field point x
effective shear force
Among four equations, any two equations can be
adopted to solve the problem.
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Expansion
Degenerate kernel (separate form)
Fourier series expansions of boundary data
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Boundary contour integration in the adaptive
observer system
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Vector decomposition
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Transformation of tensor components
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Linear system
where H denotes the number of circular boundaries
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Techniques for solving scattering problems
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Outlines
1. Introduction
2. Methods of solution
3. Illustrated examples
4. Concluding remarks
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Case 1 An infinite plate with one hole
Geometric data a 1m thickness0.002m Boundary
condition Inner edge free
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Distribution of DMCF on the circular boundary by
using different methods, the present method,
analytical solution and FEM
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Case 2 An infinite plate with two holes
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Distribution of DMCF on the circular boundary by
using different methods, the present method and
FEM
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Outlines
1. Introduction
2. Methods of solution
3. Illustrated examples
4. Concluding remarks
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Concluding remarks
A semi-analytical approach to solve the
scattering problem of flexural waves and to
determine DMCF in an infinite thin plate with
multiple circular holes was proposed
1.
The present method used the null BIEs in
conjugation with the degenerate kernels, and the
Fourier series in the adaptive observer system.
2.
The improper integrals in the direct BIEs were
avoided by employing the degenerate kernels and
were easily calculated through the series sum.
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Numerical results show that the closer the
central distance is, the larger the DMCF is.
4.
The DMCFs have been solved by using the present
method in comparison with the available exact
solutions and FEM results using ABAQUS.
5.
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The End
Thanks for your kind attention